WEBVTT

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[Music]

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Renee Sherry: Maybe we could
talk a little bit today about,

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"What do we see as the
learning trajectory

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when it comes to fractions?"

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From the very beginning
in Kindergarten,

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or even before they
come to Kindergarten,

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up through operations
and proportions.

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Ken Jensen: What's the
tip of the iceberg?

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Sherry: Well, it
seems to me operations

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with fractions would be the
tip because that's what I see

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if people are rushing anywhere,

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they're rushing to
the operation.

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And that's what I
see as the top.

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Jensen: So by this you
mean adding, subtracting.

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Sherry: Multiplying
and dividing fractions.

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Jensen: So all the
calculations working

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with those kinds of fractions.

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Sherry: Mixed, yeah.

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Jensen: So working
with fractions.

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Okay. Kim Pippenger: So you
want to put an example up?

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Jensen: Sure, so maybe
3/4 plus 5/8 and thinking

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about what they would do with
the denominator in something

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like that and what
they might need

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to be able to make sense of.

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I mean, we talk about
needing a common denominator,

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but do we talk about why you
need a common denominator?

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That might be something
that goes back

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down into this part down here.

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So I'm wondering, what kinds of
things do go towards the bottom

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of this iceberg in terms of
getting students to be able

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to make sense of what this
means so they can make sense

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of the answer when they're done?

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Pippenger: So I think
they come in with a lot

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of those informal experiences.

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I mean, even like just
cutting a sandwich in half

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and having a sense
of if the halves

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or relatively equal or not.

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Sherry: And I think
even beyond halves.

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I mean, if you have more than
one brother, and you have

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to share your cookie,

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they can...I've seen kids
even be able to do thirds

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and fourths, but
knowing the concept

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that they want the pieces
to be the same size.

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Jensen: How can we build
their understanding,

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so make that connection between
what kids are really coming

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with, what they already
know, and where we need

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to get them, say,
by third grade?

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Sherry: I think we need

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to probably do a
little bit more work

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about having younger kids
find fractions of sets,

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and maybe not even using that
word, but if these are all M&M's

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and we have to split
them between us,

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that's also partitioning.

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Pippenger: Yeah,
that sharing piece.

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Sherry: And I'm not sure
we always hit on that

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as much as we possibly could.

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As I'm thinking back along
the iceberg, I'm thinking

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that the preformal kids are
coming with some knowledge

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around splitting things
into equal pieces.

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Jensen: And they're
coming in really

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with all three models
that you can access.

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Sherry: Right.

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Jensen: We've got to help
teachers access what kids are

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coming with.

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Sherry: Right.

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Pippenger: I think what you're
getting at with the models, too,

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is that different
models are going

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to help kids see different
things, and so it's possible

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that even though I may have some
of these earlier experiences,

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when you give me sort
of a different model

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or a different context,
I kind of go back

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to the beginning again.

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And that gets to that decision
making that we want kids to have

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about which model
is going to help me,

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when is the number
line going to help me,

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when is the array
model going to help me,

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when is the pattern
block going to help me.

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So how do we start building that
decision making with kids, too?

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Sherry: I think the next one is
comparing-which one's bigger,

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which one's smaller-and
then being able

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to order a set in
the trajectory.

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Pippenger: I think, too,
as much as we want learning

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to be linear, it's not.

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I mean, kids, especially
tangled kids

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that are a little bit older,
tend to have pieces from higher

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up on the iceberg and
then missing pieces

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from lower on the iceberg.

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I mean, not always, so I
think we can't get too locked

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into first they do
this, then they do this.

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Sherry: I mean, after I
think of order and being able

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to compare two fractions
from the same whole,

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I think what I think
about next as far

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as the learning trajectory
would be equivalent fractions.

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There's a reason we find
equivalent fractions,

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and that's for the use for
further on in operations

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or for whatever the use is.

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Have we helped our
teachers understand

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that it's not just an exercise
in finding equivalent fractions,

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that's there's a use
and a purpose for it?

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It has to do with
teacher confidence.

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Pippenger: And I think
it has to do with knowing

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that progression as well.

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So knowing my kids have
come from some place,

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and if they're not right here
with me, I know I can go back

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and I can kind of fill
in some gaps for them.

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And then I also know that
this math is going some

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place important.

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Jensen: That leads into
where I think the power

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of the iceberg really comes in.

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I can think about a
teacher who wants to get

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at this particular
concept [in] sixth grade.

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This is the sixth-grade
activity right here,

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and if the only thing
the teacher says is

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"Find a common denominator,
multiply both-the three

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and the four by two- and go,"

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and the teacher doesn't
understand all these things

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that are down here, they
don't have anything to pull

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from to make sense of this.

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[Music]